Predicting the Most Recent Number of Infected Individuals in Different Countries
Catherine Al Aswad, Ali Alhakeem, Uyen Dao, Long Kim Long
Last Updated 11/17/2020
I. Introduction
Table 1.Sample for 5 randomly chosen countries of the data set used in this study| Country | cumulative_confirmed_cases_per_10000 | Stringency_Index | Economic_Support_Index | Economic_Support_Index_levels |
|---|---|---|---|---|
| Poland | 60.129772 | 50.93 | 37.5 | [37.5,50) |
| Bangladesh | 24.312931 | 62.96 | 50.0 | [50,62.5) |
| Togo | 2.674959 | 53.70 | 50.0 | [50,62.5) |
| Dominican Republic | 114.417991 | 78.70 | 25.0 | [25,37.5) |
| Mozambique | 3.868796 | 62.04 | 0.0 | [0,12.5) |
| Country | Population2019 | age15_64_population_prop_2019 | nurses_midwives_per_1000_2018 | Smoking_prevalence_15_2016 |
|---|---|---|---|---|
| Poland | 37970874 | 66.69772 | 6.8926 | 28.0 |
| Bangladesh | 163046161 | 67.60528 | 0.4124 | 23.0 |
| Togo | 8082366 | 56.11430 | 0.4102 | 7.4 |
| Dominican Republic | 10738958 | 64.99292 | 1.3802 | 13.7 |
| Mozambique | 30366036 | 52.75355 | 0.6847 | 16.6 |
II. Exploratory data analysis
Table 2: Summary for the cumulative confirmed cases per 10,000
| n | min | median | mean | max | sd |
|---|---|---|---|---|---|
| 78 | 0.0334753 | 27.50465 | 67.20095 | 461.5392 | 92.11358 |
Figure 1. Distribution for the cumulative confirmed cases per 10,000 for individual countries
Figure 2. Distribution for the government response measured by the Stringency Index
Figure 3. Distribution for the government response measured by the Economic Support Index
Figure 4. Distribution for the Proportion of population that is 15-64 years old, in 2019 for individual countries
Figure 5. Distribution for nurses and midwives per 1000 in 2018 for individual countries
Figure 6. Distribution for the Smoking Prevalence for 15+ years olds, in 2016 for individual countries
Figure 7.1. Interactive Scatterplot for the cumulative confirmed cases per 10,000 for individual countries against their government response measured by the Stringency Index. The red line is the best fit line. The blue curve is the Loess curve. The vertical black lines indicate the chosen knot locations at the 5th percentile SI = 36.1100, 35th percentile SI = 61.0640, 65th percentile SI= 77.3335, 95th percentile SI= 92.7295
Figure 7.2. Interactive Scatterplot for the cumulative confirmed cases per 10,000 for individual countries against their government response measured by the Stringency Index, grouped by the Economic Support Index levels
Figure 8. Interactive Scatterplot for the cumulative confirmed cases per 10,000 for individual countries against their government response measured by the Economic Support Index. The red line is the best fit line. The blue curve is the Loess curve.
Figure 9. Interactive Scatterplot for the cumulative confirmed cases per 10,000 for individual countries against their Proportion of population that is 15-64 years old, in 2019. The red line is the best fit line. The blue curve is the Loess curve. The vertical black lines indicate the chosen knot locations at the 5th percentile APP = 53.54730, 35th percentile APP = 62.88236, 65th percentile APP = 65.88236, 95th percentile APP = 72.63432.
Technically, could not find an article that proves it is relevant, even though we think there may be some relevance, so no splines here.
Figure 10. Interactive Scatterplot for the cumulative confirmed cases per 10,000 for individual countries against their Smoking prevalence for 15+ year olds in 2016. The red line is the best fit line. The blue curve is the Loess curve.
Figure 11. Interactive Scatterplot for the cumulative confirmed cases per 10,000 for individual countries against their Service coverage index in 2017. The red line is the best fit line. The blue curve is the Loess curve.The vertical black lines indicate the chosen knot locations at the 5th percentile NM = 0.434670, 35th percentile NM = 1.654765, 65th percentile NM = 4.196680, 95th percentile NM = 12.579690.
Figure 12. Boxplot of relationship between the cumulative confirmed cases per 10,000 for individual countries and the Economic Support Index levels
III. Multiple linear regression
i. Methods
Figure 13. Distribution for the cumulative confirmed cases per 10,000 raised to 0.5, for individual countries
Using the following model:
## lm(formula = cumulative_confirmed_cases_per_10000_transf ~ ns(Stringency_Index,
## knots = c(36.11, 61.064, 77.3335, 92.7295)) + Economic_Support_Index +
## ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236,
## 65.88236, 72.63432)) + ns(nurses_midwives_per_1000_2018,
## knots = c(0.43467, 1.654765, 4.19668, 12.57969)) + Smoking_prevalence_15_2016,
## data = tidy_joined_dataset)
Figure 14. Normal Q-Qplot for the cumulative number of confirmed cases per 10000, raised to 0.5
Figure 15. Residuals distribution for the statistical model
Figure 16. Residuals graph for the fitted values, with a Lowess curve in blue and a horizontal line at zero in red.
Figure 17. Residuals graph for the Stringency Index, with a Lowess curve in blue and a horizontal line at zero in red.
Figure 18. Residuals graph for the Economic Support Index, with a Lowess curve in blue and a horizontal line at zero in red.
Figure 19. Residuals graph for the Proportion of population that is 15-64 years old, in 2019, with a Lowess curve in blue and a horizontal line at zero in red.
Figure 20. Residuals graph for the Smoking prevalence for 15+ year olds in 2016, with a Lowess curve in blue and a horizontal line at zero in red.
Figure 21. Residuals graph for the nurses and midwives per 1000 in 2018, with a Lowess curve in blue and a horizontal line at zero in red.
Table 3: VIF table
| GVIF | Df | GVIF^(1/(2*Df)) | |
|---|---|---|---|
| ns(Stringency_Index, knots = c(36.11, 61.064, 77.3335, 92.7295)) | 3.249592 | 5 | 1.125079 |
| Economic_Support_Index | 1.349972 | 1 | 1.161883 |
| ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236, 65.88236, 72.63432)) | 3.108132 | 5 | 1.120082 |
| ns(nurses_midwives_per_1000_2018, knots = c(0.43467, 1.654765, 4.19668, 12.57969)) | 5.211657 | 5 | 1.179499 |
| Smoking_prevalence_15_2016 | 1.440917 | 1 | 1.200382 |
ii. Model Results and Interpretation
Table 4. Model Summary Table
## lm(formula = cumulative_confirmed_cases_per_10000_transf ~ ns(Stringency_Index,
## knots = c(36.11, 61.064, 77.3335, 92.7295)) + Economic_Support_Index +
## ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236,
## 65.88236, 72.63432)) + ns(nurses_midwives_per_1000_2018,
## knots = c(0.43467, 1.654765, 4.19668, 12.57969)) + Smoking_prevalence_15_2016,
## data = tidy_joined_dataset)| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | -10.4612 | 4.7789 | -2.1890 | 0.0325 |
| ns(Stringency_Index, knots = c(36.11, 61.064, 77.3335, 92.7295))1 | 10.1282 | 3.6028 | 2.8112 | 0.0067 |
| ns(Stringency_Index, knots = c(36.11, 61.064, 77.3335, 92.7295))2 | 9.3930 | 4.0014 | 2.3474 | 0.0222 |
| ns(Stringency_Index, knots = c(36.11, 61.064, 77.3335, 92.7295))3 | 17.0716 | 3.5971 | 4.7459 | 0.0000 |
| ns(Stringency_Index, knots = c(36.11, 61.064, 77.3335, 92.7295))4 | 16.2272 | 7.7118 | 2.1042 | 0.0396 |
| ns(Stringency_Index, knots = c(36.11, 61.064, 77.3335, 92.7295))5 | 0.7466 | 3.0984 | 0.2410 | 0.8104 |
| Economic_Support_Index | 0.0257 | 0.0175 | 1.4694 | 0.1470 |
| ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236, 65.88236, 72.63432))1 | 1.1669 | 3.1130 | 0.3748 | 0.7091 |
| ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236, 65.88236, 72.63432))2 | 6.4597 | 3.4426 | 1.8764 | 0.0655 |
| ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236, 65.88236, 72.63432))3 | 2.8432 | 3.7089 | 0.7666 | 0.4463 |
| ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236, 65.88236, 72.63432))4 | 11.7261 | 6.7166 | 1.7458 | 0.0860 |
| ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236, 65.88236, 72.63432))5 | 12.4694 | 3.2654 | 3.8186 | 0.0003 |
| ns(nurses_midwives_per_1000_2018, knots = c(0.43467, 1.654765, 4.19668, 12.57969))1 | 1.7848 | 2.2335 | 0.7991 | 0.4274 |
| ns(nurses_midwives_per_1000_2018, knots = c(0.43467, 1.654765, 4.19668, 12.57969))2 | -0.0042 | 3.4184 | -0.0012 | 0.9990 |
| ns(nurses_midwives_per_1000_2018, knots = c(0.43467, 1.654765, 4.19668, 12.57969))3 | 9.8196 | 3.7608 | 2.6110 | 0.0114 |
| ns(nurses_midwives_per_1000_2018, knots = c(0.43467, 1.654765, 4.19668, 12.57969))4 | 8.7379 | 4.2247 | 2.0683 | 0.0429 |
| ns(nurses_midwives_per_1000_2018, knots = c(0.43467, 1.654765, 4.19668, 12.57969))5 | 8.2892 | 2.9914 | 2.7710 | 0.0074 |
| Smoking_prevalence_15_2016 | -0.0427 | 0.0498 | -0.8574 | 0.3946 |
| Value | df | |
|---|---|---|
| Residual Standard Error | 3.467 | 60 |
| Multiple R-squared | 0.643 | |
| Adjusted R-squared | 0.542 |
| Value | Numerator df | Denominator df | |
|---|---|---|---|
| Model F-statistic | 6.363 | 17 | 60 |
| P-value | 3.565e-08 |
\[\begin{aligned} H_0:&\beta_{SI, 0-25} = 0 \\\ \mbox{vs }H_A:& \beta_{SI, 0-25} \neq 0 \end{aligned}\] \[\begin{aligned} H_0:&\beta_{SI, 25-50} = 0 \\\ \mbox{vs }H_A:& \beta_{SI, 25-50} \neq 0 \end{aligned}\] \[\begin{aligned} H_0:&\beta_{SI, 50-75} = 0 \\\ \mbox{vs }H_A:& \beta_{SI,50-75} \neq 0 \end{aligned}\] \[\begin{aligned} H_0:&\beta_{SI, 75-100} = 0 \\\ \mbox{vs }H_A:& \beta_{SI, 75-100} \neq 0 \end{aligned}\]
\[\begin{aligned} H_0:&\beta_{15to65 APP, 50-67.5} = 0 \\\ \mbox{vs }H_A:& \beta_{15to65 APP, 50-67.5} \neq 0 \end{aligned}\] \[\begin{aligned} H_0:&\beta_{15to65 APP, 67.5-80} = 0 \\\ \mbox{vs }H_A:& \beta_{15to65 APP, 67.5-80} \neq 0 \end{aligned}\] \[\begin{aligned} H_0:&\beta_{NM, 0-10} = 0 \\\ \mbox{vs }H_A:& \beta_{NM, 0-10} \neq 0 \end{aligned}\] \[\begin{aligned} H_0:&\beta_{NM, 10-20} = 0 \\\ \mbox{vs }H_A:& \beta_{NM, 10-20} \neq 0 \end{aligned}\]
\[\begin{aligned} H_0:&\beta_{ESI} = 0 \\\ \mbox{vs }H_A:& \beta_{ESI} \neq 0 \end{aligned}\] \[\begin{aligned} H_0:&\beta_{SP} = 0 \\\ \mbox{vs }H_A:& \beta_{SP} \neq 0 \end{aligned}\]
iii. Inference for multiple regression
Table 5. ANOVA Table| Df | Sum Sq | Mean Sq | F value | Pr(>F) | |
|---|---|---|---|---|---|
| ns(Stringency_Index, knots = c(36.11, 61.064, 77.3335, 92.7295)) | 5 | 390.6686 | 78.1337 | 6.4999 | 0.0001 |
| Economic_Support_Index | 1 | 126.2974 | 126.2974 | 10.5065 | 0.0019 |
| ns(age15_64_population_prop_2019, knots = c(53.5473, 62.88236, 65.88236, 72.63432)) | 5 | 531.3317 | 106.2663 | 8.8402 | 0.0000 |
| ns(nurses_midwives_per_1000_2018, knots = c(0.43467, 1.654765, 4.19668, 12.57969)) | 5 | 243.1708 | 48.6342 | 4.0458 | 0.0031 |
| Smoking_prevalence_15_2016 | 1 | 8.8368 | 8.8368 | 0.7351 | 0.3946 |
| Residuals | 60 | 721.2501 | 12.0208 | NA | NA |
Figure 22. Interactive Scatterplot for the cumulative confirmed cases per 10,000 (raised to 0.5) for individual countries against their government response measured by the Stringency Index, where median economic support index = 50, median population proportion of ages 15 to 64 in 2019 = 64.65951, median nurses midwives per 1000 in 2018 = 2.47285, and median Smoking prevalence for people ages 15+ in 2016 = 17.05. The blue line is the spline, with its associated 95% CI and wider pink 95% PI. The vertical black lines indicate the chosen knot locations at the 5th percentile SI = 36.1100, 35th percentile SI = 61.0640, 65th percentile SI= 77.3335, 95th percentile SI= 92.7295.
Figure 23. Interactive Scatterplot for the cumulative confirmed cases per 10,000 (raised to 0.5) for individual countries against their nurses and midwives per 1000 in 2018, where median Stringency index = 70.14, median economic support index = 50, median population proportion of ages 15 to 64 in 2019 = 64.65951, and median Smoking prevalence for people ages 15+ in 2016 = 17.05. The blue line is the spline, with its associated 95% CI and wider pink 95% PI. The vertical black lines indicate the chosen knot locations at the 5th percentile NM = 0.434670, 35th percentile NM = 1.654765, 65th percentile NM = 4.196680, 95th percentile NM = 12.579690.
Figure 24. Interactive Scatterplot for the cumulative confirmed cases per 10,000 (raised to 0.5) for individual countries against their Proportion of population that is 15-64 years old, in 2019, where median Stringency index = 70.14, median economic support index = 50, median nurses midwives per 1000 in 2018 = 2.47285, and median Smoking prevalence for people ages 15+ in 2016 = 17.05. The blue line is the spline, with its associated 95% CI and wider pink 95% PI.The vertical black lines indicate the chosen knot locations at the 5th percentile APP = 53.54730, 35th percentile APP = 62.88236, 65th percentile APP = 65.88236, 95th percentile APP = 72.63432.
Figure 25. Interactive Scatterplot for the cumulative confirmed cases per 10,000 (raised to 0.5) for individual countries against their Smoking prevalence for 15+ year olds in 2016, where median Stringency index = 70.14, median economic support index = 50, median population proportion of ages 15 to 64 in 2019 = 64.65951, and median nurses midwives per 1000 in 2018 = 2.47285. The blue line is the spline, with its associated 95% CI and wider pink 95% PI.
Figure 26. Interactive Scatterplot for the cumulative confirmed cases per 10,000 (raised to 0.5) for individual countries against their government response measured by the Economic Support Index, where median Stringency index = 70.14, median population proportion of ages 15 to 64 in 2019 = 64.65951, median nurses midwives per 1000 in 2018 = 2.47285, and median Smoking prevalence for people ages 15+ in 2016 = 17.05. The blue line is the spline, with its associated 95% CI and wider pink 95% PI.
If I transformed them back, then the intervals will not be symmetric.
Table 6. The 95% Prediction intervals for the \(\text{cumulative confirmed cases per 10,000}^{0.5}\), where Stringency Index = 20, 50, 70.14, 90, respectively, for median economic support index = 50, median population proportion of ages 15 to 64 in 2019 = 64.65951, median nurses midwives per 1000 in 2018 = 2.47285, and median Smoking prevalence for people ages 15+ in 2016 = 17.05.| SI | Point Estimate | Lower Limit | Upper Limit |
|---|---|---|---|
| 30.00 | -0.52479 | -8.48049 | 7.43092 |
| 50.00 | 4.94393 | -2.50340 | 12.39126 |
| 70.14 | 5.74856 | -1.50817 | 13.00530 |
| 90.00 | 10.64518 | 3.13071 | 18.15964 |